Question

PLEASE HELP on these problems! I really need the help and these are due in a few hours. Very urgent. I would really appreciate the help, so please help!!!
1. If f(x) = a(r)^x is an example of exponential decay, what must be the value of a and r?

2. The population of Greenfield was 52,500 at the beginning of 1980. Its population steadily decreased by 2.5% per year from 1980 through 1990. What is the population of Greenfield at the end of 1990?

3. If g(x) is the inverse of f(x) - 2x + 1, find a point on g(x) for which both coordinates are positive integers less than 10.

Answer 1

For 1, since the exponential decay formula is a*(1-r)^x, a is still the same as the base value and r would replace 1-r with r being the decay rate, so r would be 1-the decay rate.

For 2, we apply that formula, getting 52500*(1-2.5%)^10 (because it's for 10 years and compounded annually). Since 100% is 1, 2.5% is 0.025, and
1-0.025=0.975. Plugging that in, we get 52500*(0.975)^10=around 40757 people.

For inverse functions, what we do is have y=2x+1, so we switch x and y and solve for y, so in x=2y+1 we subtract one from both sides to get x-1=2y. Next, dividing both sides by 2, we get (x-1)/2=y as our inverse. Since we want positive integers less than 10, (x-1) has to be even and therefore x has to be odd. Using a bit of guess and check, we get that x can be 9 while y can be 4 for (9, 4)